AdvancedTopic 92 of 104Engineering Mathematics
Interpolation
Estimates values between known data points using polynomials or splines.
In depth
Interpolation constructs a function through known points. Linear uses straight segments, Lagrange or Newton polynomials fit all points (but oscillate at high degree), and cubic splines give smooth piecewise curves. Used for table lookups and resampling.
Key points
- Splines avoid Runge oscillation
- Interpolation ≠ extrapolation
- Error depends on spacing
ExampleSteam table values between 100 °C and 110 °C are linearly interpolated.
Interpolation estimates values between measurements, so it is used for tables, calibration curves and sampled data.
What is Interpolation?
Estimates values between known data points using polynomials or splines.
Why is it important to understand interpolation?
Interpolation estimates values between measurements, so it is used for tables, calibration curves and sampled data.
What is the difference between interpolation and extrapolation?
Interpolation estimates within the data range and is generally reliable. Extrapolation goes outside the range and can be very wrong.
Why use splines instead of a single polynomial?
High-degree polynomials oscillate between points. Splines use low-order pieces joined smoothly and avoid that problem.
How is Interpolation used in practice?
Typical applications include converting a steam table value between two tabulated pressures, estimating a sensor reading between two calibration points and filling in missing samples in a time series.
What should I learn before Interpolation?
Study "Root-finding methods" first (topic 91 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Interpolation?
Splines avoid Runge oscillation. Interpolation ≠ extrapolation. Error depends on spacing.
Can you give an example of Interpolation?
Steam table values between 100 °C and 110 °C are linearly interpolated.
What comes after Interpolation?
The next topic is "Numerical differentiation", which builds directly on this one.
How difficult is Interpolation?
It is classed as Advanced in Engineering Mathematics (topic 92 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.